Figure 2: Distribution array of coherent states at a G′-C′ (symmetric) or *G-*C (asymmetric) site. Symmetric, asymmetric and second asymmetric (unlabeled) channels (→) by which metastable keto‑amino G‑C protons populate enol‑imine states. Dashed arrows identify pathways for quantum mechanical flip‑flop of enol‑imine protons. Approximate electronic structures for hydrogen bond end groups and corresponding proton positions are shown for the metastable keto‑amino duplex (a) and for enol‑imine G'-C' coherent states (b-e). Electron lone‑pairs are represented by double dots, :, and a proton by a circled H. Proton states are specified by a compact notation, using letters G, C, A, T for DNA bases with 2’s and 0’s identifying electron lone-pairs and protons, respectively, donated to the hydrogen bond by – from left to right – the 6‑carbon side chain, the ring nitrogen and the 2‑carbon side chain. Superscripts identify the component at the outside position (in major and minor groves) as either an amino group proton, designated by 00, or a keto group electron lone‑pair, indicated by 22. Superscripts are suppressed for enol and imine groups.

Hydrogen bonds in duplex DNA genomes are replicated into the metastable keto-amino state where reduced
energy, enol and imine proton qubit states are initially unoccupied, but are energetically accessible via EPR
isomerization [1-11,27-30]. In the asymmetric case, quantum uncertainty limits, Δx Δpx ≥ ћ/2, operate on hydrogen
bonding amino (−NH2) protons of cytosine, causing confinement of amino protons to too small of space, Δx [75]. This
creates direct quantum mechanical proton – proton physical interaction, which generates the asymmetric EPR
arrangement, keto-amino → enol-imine (Figure 1b), where position and momentum entanglement is introduced
between separating imine and enol protons. Molecular neutrality and stability of complementary *G-*C quantum
entanglement states within the double helix are satisfied by transfer of an amino cytosine proton to an electron
lone-pair on the complementary guanine keto oxygen, and transfer of the hydrogen bonded ring proton, from
guanine to cytosine, and simultaneously, intra molecular reorganization of appropriate π and σ electrons, illustrated
in Figure 1b. Each reduced energy, entangled imine and enol product proton is shared between two
indistinguishable sets of electron lone-pairs, and therefore, participates in entangled quantum oscillations [1-9] at ~
1013 s−1 between near symmetric energy wells in decoherence-free subspaces [37,38]. This specifies quantum
dynamics of an EPR pair of entangled enol and imine protons until measured by an enzyme quantum processor.
The imine and enol protons constitute a pair of entangled two-state proton qubit on opposite DNA strands.

An entangled imine or enol proton is in state │+ > when it is in position to participate in inter strand hydrogen
bonding and is in state │− > when it is “outside”, in a major or minor DNA groove [44-46,103].The quantum mechanical
state of the entangled pair of proton qubit can be viewed as a vector in the four-dimensional Hilbert space that
describes the quantum position state of two protons. The most general quantum mechanical state of these two
protons can be written as

(1)

where the first symbol, + or −, represents proton 1 and the second symbol represents proton 2,and the expansion
coefficients, c’s, satisfy normalization, Since Equation (1) cannot be expressed as a tensor product of protons 1 and 2, maximally entangled quantum states for the qubit pair of imine
and enol protons can be written in terms of the four Bell [31-32] states, expressed as

(2)

(3)

(4)

(5)

Quantum informational content resides within the entangled pair of proton qubit occupying intra molecular
decoherence-free subspaces at enol and imine groups on opposite DNA strands. Due to entanglement imposed by
quantum uncertainty limits, Δx Δpx ≥ ћ/2, on metastable amino protons, “measurement” of either position or
momentum for one enol or imine proton would specify, instantaneously, the position or momentum of the other
proton on the opposite DNA strand. An entangled *C-imine proton is in state │− > when it is “outside”, in a major or
minor DNA groove (illustrated by state *C2 0 22 in Figure 2f), and is in state │+ > when it is “inside”, in position to
participate in inter strand hydrogen bonding (*C0 0 22 in Figure 2g). When *C is on the transcribed strand, the
enzyme quantum processor can “trap” the *C-imine proton in a DNA groove position in an interval, δt<< 10−13 s,
which specifies its state as │− >*C. Due to entanglement imposed by quantum uncertainty limits, Δx Δpx ≥ ћ/2, the
*G-enol proton is in state │+ >*G (*G0 2 00 in Figure 2f) on the opposite DNA strand. This quantum entanglement
requirement is in agreement with observation [1-3,7-9]. For example, in the case of T4 phage mutant, rX655UGA
(Table 1, ref. 6), *C is on the transcribed strand and *G is on the complementary strand; so, in an interval, δt<<
10−13 s, an enzyme – proton entanglement is formed between the quantum reader and the *C-imine proton
occupying a DNA groove, in state │− >*C. Before proton decoherence, τD< 10−13 s, the enzyme entangled *C2 0 22 is deciphered and transcriptionally expressed as normal T220 22 [1-3,7-9], and subsequently, enzyme quantum
coherence implements a quantum search, Δt′ ≤ 10−14 s, that selects an incoming amino proton of adenine to create
the complementary mis pair, A00 2 # - *C2 0 22 (Table 1). This is replicated to finalize the particular ts, *C2 0 22 →
T22 0 22.

The dimensionality of the Hilbert space required to express the quantum mechanical state for four proton qubit is
sixteen, i.e., 2N =24 = 16. Each entangled imine and enol proton is shared between two sets of indistinguishable
electron lone-pairs, and thus, participates in entangled quantum oscillations between near symmetric energy wells
at ~ 1013 s−1 in decoherence-free subspaces, which specifies entangled proton qubit dynamics occupying a heteroduplex heterozygote G′-C′ superposition site [7-9,35,36-39]. In this case, two sets of entangled imine and enol
proton qubit - four protons constituting two sets of entangled “qubit pairs” - occupy the complementary G′-C′
superposition isomers such that enzyme quantum reader “measurement” of G′-protons specifies, instantaneously
[27-30],quantum states of the four entangled qubit that occupy the sixteen-dimensional space.

Studies of heteroduplex heterozygote G′-C′ sites with G′ on the transcribed strand require the enzyme quantum
reader to specify and execute quantum informational content of four different entangled G′-proton configurations
(Figure 2). In the case of G′0 0 2 (G′0 0 2 → C), the carbon-2 imine proton is in state │− > groove position, whereas
the Eigen state G′2 0 2 (G′2 0 2 → T) has both carbon-2 imine and carbon-6 enol protons in state │− > groove
positions. Eigen state G′2 0 0 (G′2 0 0 → G; “null” mutation) has the carbon-6 enol proton “trapped” in a state │− >
DNA groove, but entangled enol and imine protons for Eigen state G′0 0 0 are both in state │+ >, the “interior” inter
strand hydrogen bond position. Since the enol and imine quantum protons on G′ are one-half of the four entangled
imine and enol G′-C′ proton qubit pairs, enzyme quantum reader measurements on G′-proton states specifically
selectquantum mechanical qubit states, │− > and │+ >, for the four entangled G′-C′ protons. Here the entangled
pair - guanine carbon-2 imine and cytosine carbon-2 enol - is identified, respectively, as protons number I and II
(Roman numerals). Proton numbers III and IV, respectively, are cytosine carbon-6 imine and guanine carbon-6 enol.
Using this notation, the enzyme quantum reader measures the four entangled proton qubit states of G′0 0 2 as
│−+−+ >, i.e., guanine imine proton I is in state │− >, cytosine enol proton II is in state │+ >, cytosine imine proton
III is in state │− >, and guanine enol proton IV is in state │+ >. Similarly, the measured proton qubit state for G′2 0
2 is │−++− >, and is │+−+− > for G′2 0 0, and finally, is │+−−+ > for Eigen state G′0 0 0. In addition to the four
quantum mechanical states of G′ imposed by enzyme quantum reader measurements (Figure 2b-e), twelve
additional states are required to specify the four two-state quantum mechanical proton qubit. The G′-C′ site
superposition consist of two sets of intra molecular entangled proton qubit pairs that are participating in quantum
oscillations between near symmetric energy wells in decoherence-free subspaces [38-40] at ~1013 s−1 s. Therefore,
the most general quantum mechanical state of these four G′-C′ protons is given by

(6)

Where the ci’s represent, generally complex, expansion coefficients. Since the 16-state superposition of four
entangled proton qubit occupy enol and imine “intra-atomic” subspaces, shared between two indistinguishable sets
of electron lone pairs, the entangled quantum superposition system will persist in evolutionarily selected
decoherence-free subspaces until an invasive perturbation, e.g., “measurement”, exposes the previously “undisturbed” quantum mechanical superposition [1-3,40,74].Just before enzyme quantum reader measurement of a G′-C′ site where G′ is on the transcribed strand, the 16-state G′-C′ (Figure 5).

Superposition system is described by Equation (6). In an interval δt<< 10−13 s, the enzyme quantum reader simultaneously detects entangled G′-protons I (carbon-2 imine) and IV (carbon-6 enol) in either correlated position states, │−> or │+>,which are components of an entangled proton “qubit pair”. When proton I or IV is measured by the quantum reader in position state, │−> or │+>, the other member of this entangled pair will, instantaneously [27-30], be in the appropriately correlated state, │+> or │−>, respectively. Protons detected in state │−>, “outside” groove position, form “new” entanglement states with the proximal quantum reader that enable enzyme quantum coherence to implement its quantum search, Δt′ ≤ 10−14 s, which specifies an incoming electron lone-pair, or amino proton, belonging to the tautomer selected for creating the “correct” complementary mis pair (Figure 5). Protons detected in state │+>, “inside” hydrogen bonding position, contribute to specificity of the G′ genetic code, exemplified by both G′2 0 2 and *C2 0 22 “measured as” normal T22 0 22 (Figure 4) via quantum transcription and replication [7,8]. Since the quantum reader detects entangled G′-protons I and IV in states │−> or │+>, the “matching” correlated quantum states, │+> or │−>, of entangled C′-protons II and III were instantaneously specified. Consequently, enzyme quantum reader “measurement” on G′-protons I and IV converts, instantaneously, the 16-state quantum system of Equation (6) into the 4-state system - ć1│−+−+ >, ć5│−++− >, ć9│+−+− >, ć13│+−−+ >- listed in column B of Table 2 and illustrated in Figure 2b-e, where expansion coefficients, ći, are defined by and This result is displayed in Table 2 where column A identifies the unperturbed 16-state quantum system of Equation (6). Column B contains the distribution of │−> and │+> proton states - for G′-C′ protons: I, II, III, IV - generated instantaneously as a consequence of the quantum reader initially “measuring” quantum states of entangled G′-protons I and IV. The instantaneously generated quantum states - provide, instantaneously, specific instructions for the enzyme – proton entanglement before it embarks on its entangled enzyme quantum quest, Δt′ ≤ 10−14 s, Table 2 of selecting the particular incoming tautomer specified by molecular evolution, ts requirements [1-9]. Incoming tautomers selected by entangled enzyme quantum searches are identified in column C and resultant molecular clock substitutions, ts, are listed in column D of Table 2.

In an interval δt<< 10−13 s, the enzyme quantum processor measurement apparatus “traps” an entangled G′ imine and/or enol proton, H+, in a DNA groove, specified by state │−>, and consequently, the position state, │−> or │+>, is instantaneously specified for the four entangled G′-C′ protons: I, IV and II, III. In column A of Table 2, an entanglement state between the quantum reader and a “groove” proton is indicated by superscript, “*”, e.g., |*−+−+>, identifies G´ proton I as the enzyme – entangled “groove” proton. The “new” entanglement state between the quantum reader and the “trapped” proton enables enzyme quantum coherence to be immediately exploited in implementing an entangled enzyme quantum search, Δt′ ≤ 10−14 s, which ultimately specifies the particular ts as G′0 0 2 → C, G′2 0 2 → T or G′2 0 0 → G [6-8].The specificity of each ts is governed by the entangled enzyme quantum search selecting the correct incoming tautomers - syn-G22 2 #, syn-A00 2 #, C00 2 22- respectively, for Eigen states - G′0 0 2, G′2 0 2, G′2 0 0 - illustrated in Figure 4, Tables 1 and 2. Natural selection has exploited quantum entanglement properties of proton qubit, which allow enzyme – proton entanglement to specify and implement results of an entangled enzyme quantum search in an interval, Δt′ ≤ 10−14 s [1-9,25,26]. This mechanism implies that enzyme – proton entanglement implementaton of an enzyme quantum search would not be successful without instantaneous specification [27,28] of the four G′-C′ entangled proton qubit states determined by quantum reader “measurements” on the two G′-proton qubit, I and IV, associated with the transcribed strand.

After the enzyme quantum reader “traps” an entangled qubit, H+, in a DNA groove and before proton decoherence,
τD< 10−13 s, the quantum mechanical state for entangled proton qubit occupying the G′-C′ site is given by column B
of Table 2. The resulting observables yielded by quantum reader measurements are in terms of the four G′-C′
states, illustrated in Figure 2b-e, and listed in column B of Table 2. Thus, one can express the probability of finding
the system in each of its observable states generated by enzyme quantum reader measurement. For example, the
probability of the system being in state G′0 0 0–C′2 2 2 as assayed by enzyme quantum reader measurement is
expressed as

(7)

Similarly, the probabilities of the system being in states G′0 0 2–C′2 2 0, G′2 0 2–C′0 2 0 and G′2 0 0–C’0 2 2 are
given respectively by

(8)

(9)

(10)

Values for and can be determined from straightforward observables – G′0 0 2 → C and G′2 0 2 → T –
respectively [1,7,8]. Since the enzyme quantum reader deciphers G′2 0 0 as normal G22 0 00, the value of can be experimentally determined from clonal analysis [35]. The value of is determined from normalization, Observables yielded by enzyme measurements, e.g., and are in qualitative agreement with the distribution of G′–C′ states predicted by
Jorgensen’s model [107,108] shown in Figure 6. In particular, the relative contribution.of the “preferred” state, G′2 0 2,
is quantified by which is observed as the no. of G′2 0 2 → T events [7-9]. Observation shows that │ fold, rather than 2-fold, > which is consistent with Figure 6. Observation and Figure 6 imply that which provides the relation These values in the normalization expression
yield consistent with observation.

Figure 6: Secondary interaction model [106-107] applied to coherent superposition G'-C' and *G-*C states for
purposes of identifying relative base pairing energies. A +1 is assigned to each secondary interaction between
opposite charges and a –1 for an interaction between same sign charges, yielding a +4 for state (e) and a – 4 for
flip-flop states (c) and (f). The remaining four states – (a), (b), (d), (g) – are intermediate with base pairing energy
values of 0. The dashed lines identify intramolecular proton-proton repulsion.

Enzyme − Proton Entanglement For “Incoming” Tautomer Quantum Search

The enzyme quantum reader “measurement apparatus” patrols the double helix along major (~ 22 Å) and minor (~
12 Å) grooves [45-46,103], creating entanglement states between individual enol and imine entangled qubit “groove
protons” and proximal enzyme components. Davies [109] has noted that the polymerase protein has a mass of about
10−19 g, and a length of about 10−3 cm and travels at a speed of about 100 bp per sec., or about 10−5 cm
s−1[110,111]. The quantum reader polymerase energy source is ATP, and it maintains a reservoir of purines,
pyrimidines and nucleotides for base pairing operations. Curiously, the normal speed of the polymerase, ~ 10−5 cm
s−1, corresponds to the limiting speed allowed by the energy-time uncertainty relation for the operation of a
quantum clock. For a clock of mass m and size l, Wigner [112] found the relation

Equation (11) can be expressed in terms of a velocity inequality given by

Which, for this polymerase, yields a minimum velocity of about 10−5 cm s−1, implying the quantum reader enzyme
speed of operation can be confined by a form of quantum synchronization uncertainty [109]. The quantum reader
“measurement apparatus” has been evolutionarily selected to decipher, process and exploit informational content
within DNA base pairs composed of either (a) the classical keto-amino state, (b) undisturbed, enol and imine
entangled proton qubit states [1-9], including enzyme – proton entanglements participating in an entangled enzyme
quantum search, [25,26,40,69].

The enzyme quantum measurement-operator is identified by Μ, and operates on G′-proton states located on the
transcribed strand to yield three different entanglement states between groove protons and enzyme components.
From column B of Table 2, these enzymatic quantum “measurements”, and resulting enzyme-proton
entanglements, can be symbolically represented by

Acknowledgement

I thank Jacques Fresco for insight into catalytic site specificities of replicase and transcriptase systems. Roy Frieden provided useful comments and insight on quantum coherence and decoherence of enzyme – proton entanglements in biological systems, which are gratefully appreciated. The initial version of this paper benefited from insightful suggestions and challenges by Peggy Johnson and Pam Tipton, for which the author is grateful. I thank Peggie Price for her interest and encouragement while writing the final version of this manuscript. The author confirms that no conflicts of interest exist to declare x.

Löwdin PO. Quantum genetics and the aperiodic solid: Some aspects on the biological problems of heredity, mutations, aging and tumors in view of the quantum theory of the DNA molecule. Adv. Quantum Chem. 1965;2:213-359.

Wolf YI and Koonin EV. On the origin of the translation system and the genetic code in the RNA world by means of natural selection, exaptation, and sub functionalization. Biol Direct. 2007;2:14.

Brovarets OO, et al. Is the DPT tautomerization of the long A·G Watson-Crick DNA base mis pair a source of the adenine and guanine mutagenic tautomers? A QM and QTAIM response to the biologically important question. J Comput Chem.2014;35:451-466.